Find the Determinant [[e^(3x),e^(2x)],[3e^(3x),2e^(2x)]]

Problem

det(e^(3*x))

Solution

1. Identify the formula for the determinant of a 2×2 matrix [[a,b],[c,d]] which is a*d−b*c

2. Substitute the elements of the matrix into the formula: a=e^(3*x) b=e^(2*x) c=3*e^(3*x) and d=2*e^(2*x)

3. Set up the subtraction of the products of the diagonals.

det(e^(3*x))=(e^(3*x))*(2*e^(2*x))−(e^(2*x))*(3*e^(3*x))

4. Apply the laws of exponents, specifically e^u⋅e^v=e^(u+v) to simplify each term.

(e^(3*x))*(2*e^(2*x))=2*e^(3*x+2*x)=2*e^(5*x)

(e^(2*x))*(3*e^(3*x))=3*e^(2*x+3*x)=3*e^(5*x)

5. Subtract the simplified terms to find the final result.

2*e^(5*x)−3*e^(5*x)=−e^(5*x)

Final Answer

det(e^(3*x))=−e^(5*x)

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